Abstract: Metric Spaces. Normed Spaces Banach Spaces. Inner Product Spaces Hilbert Spaces. Fundamental Theorems for Normed and Banach Spaces. Further Applications: Banach Fixed Point Theorem. Spectral Theory of Linear Operators in Normed Spaces. Compact Linear Operators on Normed Spaces and Their Spectrum. Spectral Theory of Bounded Self--Adjoint Linear Operators. Unbounded Linear Operators in Hilbert Space. Unbounded Linear Operators in Quantum Mechanics. Appendices. References. Index.

Abstract: A formalism is presented that describes the time behavior of the spin density matrix of a nuclear spin system with arbitrary spin in terms of fictitious spin −(1/2) operators. This formalism is an extension of that used earlier for nuclei with spin I=1. For a spin system with n eigenstates we define for every pair of eigenstates ‖i〉 and ‖j〉 three operators Ii−jp, with p=x, y, and z, according to the three 2×2 Pauli matrices σx, σy, and σz. These operators together constitute a complete set of n2−1 independent Hermitian operators, and we can write the n×n density matrix and the spin Hamiltonian of the system in terms of the Ii−jp operators. The commutation relations among the operators make it possible in many cases to solve the equation of motion of the density matrix analytically. Three examples of the use of the Ii−jp operators are presented. Firstly a system of noninteracting spins with I=1 is considered. The Ii−jp operators for this case are compared with the Iq,k operator defined earlier. The cw sign...

Abstract: A necessary and sufficient condition is found for a polynomial Q of J variables to be such that Q (A 1, . . ., AJ) is a contraction whenever Aj (1 < j < J) are commuting linear operators on complex hilbert space satisfying jJ_ I A Aj < I.

Abstract: The spectral picture of an operator Pulling out direct summands The reducing essential matricial spectra of an operator Quasitriangular operators Spectral characterization of nonquasitriangular operators Approximation by nilpotent operators The Lomonosov technique A look at the invariant-subspace problem A model for quasinilpotent operators The Brown-Douglas-Fillmore theorem Bibliography.

Abstract: The object of this paper is to present a unified approach to multiparameter spectral theory of linear operators in Hilbert space. The theory is applicable to both bounded and unbounded operators and has application in the study of multiparameter spectral problems of ordinary differential operators. The main results include a Parseval equality and an eigenfunction expansion theorem.

Abstract: The idea of a linear operator or transformation in a Hilbert space ℌ (or a Banach space) is a direct generalization of the idea of a linear transformation in a finite-dimensional space. One point, however, needs emphasis (mainly because it is sometimes ignored, especially in books on quantum mechanics), namely, an operator A cannot be regarded as fully specified until its domain of definition (i.e., the set of those x in ℌ for which Ax is meaningful) has been specified; operators with different domains of definition have to be regarded as different operators. It is customary to require the domain of definition to be a linear set (manifold) in ℌ, for the obvious reason that if A is linear and Ax is defined in a set S, then Ay can be uniquely defined, by linearity, when y is any finite linear combination of elements of S. However, further extensions are not generally unique, except in special circumstances.

Abstract: The renormalization of gauge-invariant operators in Yang-Mills theories is discussed. An important property of the relevant Ward identities is used to show that the coupling of a subset of these (those which do not vanish by the equations of motion) to other operators takes a simple form. It is also shown that the renormalization constants appropriate to this set are independent of the gauge parameter. It is demonstrated that in certain important applications, for example the calculation of deep-inelastic scattering using the operator-product expansion, only the operators in this set are physically relevant.

Abstract: An asymptotic version of von Neumann's double commutant theorem is proved in which C*-algebras play the role of von Neumann algebras. This theorem is used to investigate asymptotic versions of simi- larity, reflexivity, and reductivity. It is shown that every nonseparable, norm closed, commutative, strongly reductive algebra is selfadjoint. Applications are made to the study of operators that are similar to normal (subnormal) operators. In particular, if T is similar to a normal (subnormal) operator and «■ is a representation of the C*-algebra generated by I, then ir(T) is similar to a normal (subnormal) operator. 1. Introduction. One of the reasons for the success of the theory of von Neumann algebras is J. von Neumann's double commutant theorem (46), which gives an alternate description of the weak closure of a selfadjoint algebra of operators. It is the purpose of this paper to prove an asymptotic version of the double commutant theorem that gives an alternate description of the norm closure of a selfadjoint algebra of operators. This asymptotic double commutant theorem helps to unify asymptotic versions of various operator-theoretic concepts (e.g., similarity, reflexivity, reductivity). Applications are made to the study of operators that are similar to normal (or subnormal) operators. Also a proof is given that a strongly reductive, nonseparable, commutative, norm closed algebra of operators is selfadjoint. Throughout, H denotes a separable, infinite-dimensional complex Hubert space, B(H) denotes the set of operators (bounded linear transformations) on H, and %(H) denotes the set of compact operators on H. Also § denotes a separable, nonempty subset of B(H). However, in §8 the separability assumptions on H and S will be dropped. If § Q B(H), then §* = {S*: S G S}, #"(§>) is the norm closed algebra generated by 1 and §, &W(S) is the weakly closed algebra generated by 1 and S, C*(§) is the C*-algebra generated by 1 and S, and W*(S ) is the von Neumann algebra generated by

Abstract: We consider the nonrelativistic N‐body scattering problem for a system of particles in which some subsets of the particles are identical. We demonstrate how the particle identity can be included in a general class of linear integral equations for scattering operators or components of scattering operators. The Yakubovskii, Yakubovskii–Narodestkii, Rosenberg, and Bencze–Redish–Sloan equations are included in this class. Algebraic methods are used which rely on the properties of the symmetry group of the system. Operators depending only on physically distinguishable labels are introduced and linear integral equations for them are derived. This procedure maximally reduces the number of coupled equations while retaining the connectivity properties of the original equations.

Abstract: This paper will study the weak and strong stabilizability problem of linear control systems on Hilbert space. It will be shown that sufficient conditions for stabilizability can be obtained by using a decomposition theorem in the structure theory of Hilbert space operators. The basic idea is "trivializing" the unitary "part" of a semigroup of bonded linear operators by means of a suitable "feedback" perturbation operator Controllability will not be involved in this process. However, it will be seen that further sufficient conditions as well as necessary conditions will be obtained with the aid of controllability. Extensions as well as limitatations of the familiar finite-dimensional results will also be discussed.

Abstract: Basic operator theory.- Moment theory.- Orthogonal theory.- The standard processes. Examples.- Limit theorems.- Discrete theory. Martingales. Stochastic integrals.- Multidimensional theory.- Hamiltonian processes and stochastic processes.- Concluding remarks.

Abstract: The inversion problem is solved for a wide class of singular integral operators, in particular for Wiener-Hopf operators in several variables, Mihlin-Calderon-Zygmund operators on bounded domains, and Folland-Stein operators on compact nondegenerate Cauchy-Riemann manifolds.

Abstract: Publisher Summary This chapter reviews linear maximal monotone operators and singular nonlinear integral equations of hammerstein type. In the application of the theory of monotone operators to the existence of solutions of nonlinear equations, the concept of maximal monotonicity plays a central role. It is therefore of importance to determine whether various concretely given monotone operators are maximal and to be able to generate maximal monotone operators satisfying given conditions.

Abstract: We provide a short proof of a theorem of W. B. Arveson in operator theory. The conclusion of this theorem is the same as that of the Corona Theorem but the hypotheses are operator theoretic. Our proof yields an exact value for the constant involved. We also comment on this theorem as a new approximation problem. If H ? denotes the bounded analytic functions on the unit disk D, then Carleson's Corona Theorem [2] states that given fl,... f,, fE H there exist gl... gn, H? such that Ejfjgj = 1 if and only if for some E > 0, EjIf1(z)I2 > for all z E D. Carleson also gives an upper bound for each of I g, . . ., I g&. In [1] W. B. Arveson has proved a theorem in operator theory whose conclusion is similar to that of the Corona Theorem but with different bounds on I g1j, .. . , I g j, since the hypotheses are operator theoretic. To state and prove this theorem we shall need the elementary facts about Toeplitz operators that we review below; see also [3, Chapters 6, 7]. Let L2 and L? be, respectively, the spaces of square integrable and essentially bounded functions on the unit circle with norms Il I fI and I I* Ilo. If P denotes the orthogonal projection of L2 onto the subspace of functions whose Fourier series contain only nonnegative powers of ei', then we may identify the usual Hardy space H2 with pL2 and identify H?? with H2 n L?. Every cp E L? generates a bounded operator M., on L2 by M92u = cgu for all u E L2 and for this operator IIMc1II = 119911J. Every cp C H' generates an analytic Toeplitz operator on H2 by T.1u = pu for all u E H2 and again T11,l = 1 1.. When both c and 4 are in H then T, = T7, T = Tp T.. For the particular case 4(z) = z, denote Tz by S, then T,pS = ST.p and S is an isometry, in fact a unilateral shift of multiplicity one on H2. Finally if the conjugate of T,, is T,*, then T* = PMSIH2. With these preliminaries out of the way, the result that we prove is the following. THEOREM. If fl, . .. , fn E H?, then there exist g, ... , gn E H?? satisfying jgj = 1 and such that (1) E t g}(z)j2 < 8 -2 for all z E D Received by the editors July 15, 1977. AMS (MOS) subject classifications (1970). Primary 47B35. 'This work was supported by the National Research Council Grant A-7352 and Canada Council Award W750614, while the author was a Visiting Fellow at I.A.S., the Australian National University. (? American Mathematical Society 1978

Abstract: This work is devoted to the construction of a theory of random elliptic operators; formulas are given for calculating the index of such an operator in terms of characteristic classes and the symbol of the operator. Bibliography: 15 titles.

Abstract: The closed operators in a Hilbert space H are characterized as quotients AB ~l of continuous operators on H such that the vector sum A*(H)+ B*(H) is closed. This leads to the function T(A) = A(\- A*A)~1/2, which is shown to map the strictly contractive operators on H reversibly onto the closed densely-defined operators, so as to preserve the selfadjoint and nonnegative conditions. Introduction. Suppose that (77, ") denote the usual product space with corresponding norm || • ||". By a closed operator in 77 is meant a linear function from a linear subspace of 77 into H which is closed in 77 X 77. Let % (H) denote the algebra of all continuous linear functions from 77 into 77 ("including" the complex numbers) and for each A in % (77), let A * denote the adjoint of A, let A ~ ' denote the inverse of the restriction of A to the closure of A*(H), and-in case A is nonnegative-let A~x/2 denote L41//2)_1. Note that for each A in % (77), A ~ XA is the orthogonal projection from H onto the closure of A *(H).

Abstract: The single-input discrete-time linear-quadratic optimal control problem, together with the classical time-domain and frequency domain conditions, is reviewed and treated from a novel point of view. We show that when the quadratic cost is not positive semidefinite, the problem of the boundedness of the infimum in both the zero and the free terminal state cases is related to the positivity of a self-adjoint Hilbert space operator. The structure of the spectrum of the operator in question, in both cases, is investigated, leading to a clarification of the linkage between the time-domain and the frequency-domain conditions for boundedness. More over, it is shown that the spectra of the operators in question reflect the relevant properties of the control problem.

Abstract: T. Kato's little-known generalization of a classic variational inequality for eigenvalues is extended to the case of normal operators and briefly discussed. It is usually not possible to evaluate precisely the eigenvalues of the linear operators which occur in realistic models in the physical sciences. It is thus a problem of great practical importance to have formulae for approximate evaluation of eigenvalues and for the errors of those approximations. The most important approximate formula for an eigenvalue is the Rayleigh-Ritz inequality, which gives an upper bound for the lowest eigenvalue of a selfadjoint operator. This is the prototype of a variational estimate, whereby a set of approximate eigenfunctions is guessed at and used to estimate the eigenvalues. The problem of obtaining lower bounds for the lowest eigenvalue of a selfadjoint operator is notoriously more difficult than the discovery of upper bounds, but some methods are widely known, though not so widely as the Rayleigh-Ritz inequality. The best such bound which relies only on the selfadjointness of the operator and the isolation of the lowest eigenvalue from the rest of the spectrum is due to G. Temple [7 Theorem 1]. The proofs of the Rayleigh-Ritz inequality and Temple's inequality show them to be straightforward applications of the spectral theorem [5], [6], and similar arguments can extend these inequalities to give useful estimates for any isolated eigenvalue of a selfadjoint operator-without the necessity of first estimating all the lower eigenvalues, as with the min-max principle. It is peculiar and unfortunate that more than two decades elapsed between Temple's original paper and the discovery of the generalization of Temple's inequality to arbitrary eigenvalues, and that this generalization has remained but little known for almost three more decades. In this paper Temple's inequality is generalized still further to the case of normal operators. Its use is not only for numerical computation, but also for the proofs of many abstract theorems about perturbation expansions and convergence of operator-valued functions [1], [2], [4], [5]. The classical result is Received by the editors February 18, 1977 and, in revised form, June 28, 1977. AMS (MOS) subject classifications (1970). Primary 47A10, 35P15.

Abstract: The following theorem is proved: Suppose H is a complex Hilbert space, and T: H - H is a monotonic, nonexpansive operator on H, and f G H. Define